Two-dimensional gas and vacancy solution approaches in the thermodynamic description of adsorption equilibrium

We present two approaches to description of the adsorption equilibrium in adsorbate-adsorbent systems, the treatment where the adsorbate is considered as a two-dimensional gas, and the treatment where the adsorbate and adsorbent form a solution where the adsorbate is the solute and vacancies play the role of the solvent. In the first case the application of different equations describing the state of a two-dimensional gas leads to some fundamental adsorption isotherm equations, for example, Henry, Volmer, Hill-deBoer, Fowler-Guggenheim, and Langmuir. In contrast, the application of different equations (existing in the theory of solutions) describing the activity of the solvent (e.g., the Wilson model or the model of Flory and Huggins) leads to some new adsorption equations and the assumption of the ideality of the solvent leads to the Langmuir adsorption isotherm. We present some new adsorption equations basing on the second method described above and on the Redlich-Kister and Wohl expansions. Assuming that the activity coefficients are given by the one-constant Margules equation, the two-constant Margules equation, the van Laar equation, the Wilson equation, and, finally, the Flory-Huggins equation, we derive the respective adsorption isotherm formulas. We also present the assumptions leading to the best-known adsorption equations derived applying the first method. The equation developed by Cochran et al. is also corrected, taking into account the fundamental assumptions of the Flory-Huggins theory.